On Davie's uniqueness for some degenerate SDEs

TL;DR

This paper extends Davie's uniqueness results to certain degenerate singular SDEs with Lévy noise, demonstrating strong existence, uniqueness, and pathwise uniqueness under broader conditions than previously known.

Contribution

It generalizes Davie's uniqueness to degenerate SDEs with Lévy processes, including cases with matrix drift and degenerate noise, beyond previous results with identity diffusion.

Findings

Strong existence and uniqueness are established for the considered SDEs.

Davie's pathwise uniqueness is proved under broader conditions.

Application to kinetic transport equations with added noise confirms the results.

Abstract

We consider singular SDEs like \begin{equation} \label{ss} dX_t = b(t, X_t) dt + A X_t dt + \sigma(t) d{L}_t , \;\; t \in [0,T], \;\; X_0 =x \in {\mathbb R}^n, \end{equation} where $A$ is a real $n \times n $ matrix, i.e., $A \in {{\mathbb R}}^n \otimes {{\mathbb R}}^n$, $b$ is bounded and H\"older continuous, $\sigma : [0,\infty) \to {{\mathbb R}}^n \otimes {{\mathbb R}}^d $ is a locally bounded function and $L= ({L}_t)$ is an ${\mathbb R}^d$-valued L\'evy process, $1 \le d \le n$. We show that strong existence and uniqueness together with $L^p$-Lipschitz dependence on the initial condition $x $ imply Davie's uniqueness or path by path uniqueness. This extends a result of [E. Priola, AIHP, 2018] proved when $n=d$, $A=0$ and $\sigma(t) \equiv I $. We apply the result to some singular degenerate SDEs associated to the kinetic transport operator $ \frac{1}{2} \triangle_v f + $ ${v \cdot \partial_{x}f} $ $+F(x,v)\cdot \partial_{v}f $ when $n =2d $ and $L$ is an ${{\mathbb R}}^d$-valued Wiener process. For such equations strong existence and uniqueness are known under H\"older type conditions on $b$. We show that in addition also Davie's uniqueness holds.